determine the vertex form and the maximum or minimum value of the function. f(x)=2x²+8x+3 a f(x)=2(x+4)²−13…

determine the vertex form and the maximum or minimum value of the function. f(x)=2x²+8x+3 a f(x)=2(x+4)²−13, the maximum value of f is −13 b f(x)=2(x+4)²−13, the minimum value of f is −13 c f(x)=2(x+2)²−5, the maximum value of f is −5 d f(x)=2(x+2)²−5, the minimum value of f is −5

determine the vertex form and the maximum or minimum value of the function. f(x)=2x²+8x+3 a f(x)=2(x+4)²−13, the maximum value of f is −13 b f(x)=2(x+4)²−13, the minimum value of f is −13 c f(x)=2(x+2)²−5, the maximum value of f is −5 d f(x)=2(x+2)²−5, the minimum value of f is −5

Answer

Explanation:

Step1: Complete the square

Given (f(x)=2x^{2}+8x + 3). Factor out the coefficient of (x^{2}) from the first two terms: (f(x)=2(x^{2}+4x)+3). For the quadratic (x^{2}+4x), using the formula ((a + b)^2=a^{2}+2ab + b^{2}), here (a = x) and (2ab=4x) (so (b = 2)), then (x^{2}+4x=(x + 2)^{2}-4). Substitute back: (f(x)=2[(x + 2)^{2}-4]+3). Expand: (f(x)=2(x + 2)^{2}-8 + 3). Simplify: (f(x)=2(x + 2)^{2}-5).

Step2: Determine maximum or minimum value

For a quadratic function in vertex form (y=a(x - h)^{2}+k), when (a>0), the parabola opens upward and the vertex ((h,k)) is a minimum point. Here (a = 2>0), (h=-2), (k=-5).

Answer:

D. (f(x)=2(x + 2)^{2}-5), the minimum value of (f) is (-5)